Search Results (13)

This study, within the framework of uncertainty theory, employs an uncertain differential equation with jumps to model the asset value process of a company, establishing a structured model of uncertain credit risk that incorporates jumps. This model is applied to the pricing of two types of credit derivatives, yielding pricing formulas for corporate zero-coupon bonds and Credit Default Swap (CDS). Through numerical analysis, we examine the impact of asset value volatility and jump magnitude on corporate default uncertainty, as well as the influence of jump magnitude on the pricing of zero-coupon bonds and CDS. The results indicate that an increase in volatility levels significantly enhances default uncertainty, and an expansion in the magnitude of negative jumps not only directly elevates default risk but also leads to a significant increase in the value of zero-coupon bonds and the price of CDS through a risk premium adjustment mechanism. Therefore, when assessing corporate default risk and pricing credit derivatives, the disturbance of asset value jumps must be considered a crucial factor. Full article

This paper considers the valuation of a vulnerable option when underlying stock is subject to liquidity risks. That is, it is assumed that the underlying stock is not perfectly liquid. We establish a framework where the stock price follows the stochastic volatility model and the option contains the default risk of the option issuer. In addition, we assume that liquidity risks are caused by stochastic market liquidity, and the default occurs at the first jump time of a stochastic Poisson process, which has a stochastic default intensity process consisting of both idiosyncratic and systematic components. By employing a change of measure, we derive an analytical formula for the value of a vulnerable option. Finally, we present several numerical examples to illustrate the sensitivity of significant parameters. Full article

This study analyzes the term structures of sovereign quanto credit default swap (CDS) spreads and currency options, which are driven by anticipated currency depreciation risk following sovereign credit default (Twin Ds). We develop consistent pricing models for these instruments using a jump-diffusion stochastic volatility model, which allows us to decompose the term structure into the risk components. We find a common risk factor between the intensity process of sovereign credit risk and the stochastic volatility of the exchange rate, and the depreciation risk mainly captures the dependence structure between these markets during periods of high market stress in the Eurozone countries. Depreciation risk is an important component of sovereign quanto CDS spreads and is evident in the negative slope of the volatility smile in the currency option market. Full article

In this paper, we propose an enhanced model for pricing vulnerable options. Specifically, our model assumes that parameters such as interest rates, jump intensity, and asset value volatility are governed by an observable continuous-time finite-state Markov chain. We take into account European vulnerable options that are exposed to both default risk and rare shocks from underlying and counterparty assets. We also consider stochastic default barriers driven by a regime-switching model and geometric Brownian motion, thus improving upon the assumption of fixed default barriers. The risky assets follow a related jump-diffusion process, whereas the default barriers are influenced by a geometric Brownian motion correlated with the risky assets. Within the framework of our model, we derive an explicit pricing formula for European vulnerable options. Furthermore, we conduct numerical simulations to examine the effects of default barriers and other related parameters on option prices. Our findings indicate that stochastic default barriers increase credit risk, resulting in a decrease in option prices. By considering the aforementioned factors, our research contributes to a better understanding of pricing vulnerable options in the context of counterparty credit risk in over-the-counter trading. Full article

This paper deals with a credit derivative pricing problem using the martingale approach. We generalize the conventional reduced-form credit risk model for a credit default swap market, assuming that the firms’ default intensities depend on the default states of counterparty firms and that the stochastic interest rate follows a jump-diffusion Cox–Ingersoll–Ross process. First, we derive the joint Laplace transform of the distribution of the vector process $(r_t,R_t)$ by applying piecewise deterministic Markov process theory and martingale theory. Then, using the joint Laplace transform, we obtain the explicit pricing of defaultable bonds and a credit default swap. Lastly, numerical examples are presented to illustrate the dynamic relationships between defaultable securities (defaultable bonds, credit default swap) and the maturity date. Full article

In a default event, several obligors simultaneously experience financial difficulty in servicing their debt to the point where the entire market can experience a sudden yet significant jump to a credit default. To help protect lenders against a jump-to-default event, regulators require banks to hold capital equivalent to the default risk charge as a buffer against the losses they may incur. The Basel regulatory committee has articulated and set default risk modelling guidelines to improve comparability amongst banks and enable a consistent bank-wide default risk charge estimation. Emerging markets are unique because they usually have illiquid markets and sparse data. Thus, implementing an emerging market default risk model and, at the same time, complying with the regulatory guidelines can be non-trivial. This research presents a framework for modelling the default risk charge in emerging markets in line with the regulatory requirements. The default correlation model inputs are derived and empirically calibrated using emerging market data. The paper ends with some considerations that regulators, supervisors, and banks can use to get financial institutions to adopt an emerging markets default risk charge model. Full article

Transposable elements (TEs), also known as “jumping genes”, are repetitive sequences with the capability of changing their location within the genome. They are key players in many different biological processes in health and disease. Therefore, a reliable quantification of their expression as transcriptional units is crucial to distinguish between their independent expression and the transcription of their sequences as part of canonical transcripts. TEs quantification faces difficulties of different types, the most important one being low reads mappability due to their repetitive nature preventing an unambiguous mapping of reads originating from their sequences. A large fraction of TEs fragments localizes within introns, which led to the hypothesis that intron retention (IR) can be an additional source of bias, potentially affecting accurate TEs quantification. IR occurs when introns, normally removed from the mature transcript by the splicing machinery, are maintained in mature transcripts. IR is a widespread mechanism affecting many different genes with cell type-specific patterns. We hypothesized that, in an RNA-seq experiment, reads derived from retained introns can introduce a bias in the detection of overlapping, independent TEs RNA expression. In this study we performed meta-analysis using public RNA-seq data from lymphoblastoid cell lines and show that IR can impact TEs quantification using established tools with default parameters. Reads mapped on intronic TEs were indeed associated to the expression of TEs and influence their correct quantification as independent transcriptional units. We confirmed these results using additional independent datasets, demonstrating that this bias does not appear in samples where IR is not present and that differential TEs expression does not impact on IR quantification. We concluded that IR causes the over-quantification of intronic TEs and differential IR might be confused with differential TEs expression. Our results should be taken into account for a correct quantification of TEs expression from RNA-seq data, especially in samples in which IR is abundant. Full article

The classic credit risk structured model assumes that risky asset values obey geometric Brownian motion. In reality, however, risky asset values are often not a continuous and symmetrical process, but rather they appear to jump and have asymmetric characteristics, such as higher peaks and fat tails. On the other hand, there are real Knight uncertainty risks in financial markets that cannot be measured by a single probability measure. This work examined a structural credit risk model in the Lévy market under Knight uncertainty. Using the Lévy–Laplace exponent, we established dynamic pricing models and obtained intervals of prices for default probability, stock values, and bond values of enterprise, respectively. In particular, we also proved the explicit solutions for the three value processes above when the jump process is assumed to follow a log-normal distribution. Finally, the important impacts of Knightian uncertainty on the pricing of default probability and stock values of enterprise were studied through numerical analysis. The results showed that the default probability of enterprise, the stock values, and bond values were no longer a certain value, but an interval under Knightian uncertainty. In addition, the interval changed continuously with the increase in Knightian uncertainty. This result better reflected the impact of different market sentiments on the equilibrium value of assets, and expanded decision-making flexibility for investors. Full article

Corporate bond defaults in different sectors often increase suddenly at roughly similar times, although some sectors see default rates jump earlier than others. This could reflect contagion among sectors—specifically, defaults in one sector leading to credit stresses in other sectors of the economy that would not otherwise have seen stresses. To complicate matters, simple correlation-based tests for contagion are often biased, reflecting increased volatility in periods of stress. This paper uses sectoral default data from over 30 sectors to test for signs of contagion over the past 30 years. While jumps in sectoral default rates do often coincide, there is no consistent evidence of contagion across different periods of stress from unbiased test results. Instead, coincident jumps in sectoral default rates are likely to reflect common macroeconomic shocks. Full article

Measures of corporate credit risk incorporate compensation for unpredictable future changes in the credit environment and compensation for expected default losses. Since the launch of purchases of government securities and corporate securities by the European Central Bank, it has been discussed whether the observed reduction in corporate credit risk was due to the decrease in risk aversion favored by the monetary easing or by expectations of lower losses due to corporate defaults. This work introduces a new methodology to break down the factors that drive corporate credit risk, namely the premium linked to cyclical and monetary conditions and that linked to the restructuring of the companies. Untangling these two components makes it possible to quantify the drivers of excess returns in the corporate bond market. Full article

In this paper, we study a generalised CIR process with externally-exciting and self-exciting jumps, and focus on the distributional properties and applications of this process and its aggregated process. The aim of the paper is to introduce a more general process that includes many models in the literature with self-exciting and external-exciting jumps. The first and second moments of this jump-diffusion process are used to calculate the insurance premium based on mean-variance principle. The Laplace transform of aggregated process is derived, and this leads to an application for pricing default-free bonds which could capture the impacts of both exogenous and endogenous shocks. Illustrative numerical examples and comparisons with other models are also provided. Full article

In this work, we introduce a general framework for incorporating stochastic recovery into structural models. The framework extends the approach to recovery modeling developed in Cohen and Costanzino (2015, 2017) and provides for a systematic way to include different recovery processes into a structural credit model. The key observation is a connection between the partial information gap between firm manager and the market that is captured via a distortion of the probability of default. This last feature is computed by what is essentially a Girsanov transformation and reflects untangling of the recovery process from the default probability. Our framework can be thought of as an extension of Ishizaka and Takaoka (2003) and, in the same spirit of their work, we provide several examples of the framework including bounded recovery and a jump-to-zero model. One of the nice features of our framework is that, given prices from any one-factor structural model, we provide a systematic way to compute corresponding prices with stochastic recovery. The framework also provides a way to analyze correlation between Probability of Default (PD) and Loss Given Default (LGD), and term structure of recovery rates. Full article

Default probability is a fundamental variable determining the credit worthiness of a firm and equity volatility estimation plays a key role in its evaluation. Assuming a structural credit risk modeling approach, we study the impact of choosing different non parametric equity volatility estimators on default probability evaluation, when market microstructure noise is considered. A general stochastic volatility framework with jumps for the underlying asset dynamics is defined inside a Merton-like structural model. To estimate the volatility risk component of a firm we use high-frequency equity data: market microstructure noise is introduced as a direct effect of observing noisy high-frequency equity prices. A Monte Carlo simulation analysis is conducted to (i) test the performance of alternative non-parametric equity volatility estimators in their capability of filtering out the microstructure noise and backing out the true unobservable asset volatility; (ii) study the effects of different non-parametric estimation techniques on default probability evaluation. The impact of the non-parametric volatility estimators on risk evaluation is not negligible: a sensitivity analysis defined for alternative values of the leverage parameter and average jumps size reveals that the characteristics of the dataset are crucial to determine which is the proper estimator to consider from a credit risk perspective. Full article